Finite Group DatabasebetaFinite Group DatabasebetaOrder 864.
288 conjugacy classes of subgroups; only the 109 normal subgroups are shown.
This group has 288 conjugacy classes of subgroups — too many to draw. Only its 109 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×C8×H(3,𝔽3) | C2 | semidirect |
| 3. C2×(C3⋊(C3⋊C8))⋊C3 | C2 | semidirect |
| 4. C8×(C3⋊D3)⋊C3 | C2 | direct (N × H) |
| 5. C2×C8×C3⋊D3 | C3 | semidirect |
| 12. C8×H(3,𝔽3) | C22 | semidirect |
| 13. (C3⋊(C3⋊C8))⋊C3 | C22 | semidirect |
| 17. C2×C32×C8 | D3 | semidirect |
| 18. C2×C32×C8 | C2×C3 | semidirect |
| 19. C2×C3⋊(C3⋊C8) | C2×C3 | semidirect |
| 20. C8×C3⋊D3 | C2×C3 | semidirect |
| 24. C2×(C3⋊D3)⋊C3 | C8 | direct (N × H) |
| 44. C32×C8 | C2×D3 | semidirect |
| 45. C32×C8 | C22×C3 | semidirect |
| 46. C3⋊(C3⋊C8) | C22×C3 | semidirect |
| 48. C2×H(3,𝔽3) | C2×C8 | semidirect |
| 50. (C3⋊D3)⋊C3 | C2×C8 | direct (N × H) |
| 67. C2×C3⋊D3 | C3×C8 | semidirect |
| 80. H(3,𝔽3) | C22×C8 | semidirect |
| 106. C2×C32 | D3×C8 | semidirect |
| 107. C2×C32 | C2×C3×C8 | semidirect |
| 110. C3⋊D3 | C2×C3×C8 | semidirect |
| 115. C2×C8 | (C3⋊D3)⋊C3 | direct (N × H) |
| 138. C32 | C2×D3×C8 | semidirect |
| 139. C32 | C22×C3×C8 | semidirect |
| 146. C8 | C2×(C3⋊D3)⋊C3 | direct (N × H) |
| 169. C2 | C8×(C3⋊D3)⋊C3 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C4×(C3⋊D3)⋊C3 | 2 |
| C2×C8×H(3,𝔽3) | 2 |
| C2×(C3⋊(C3⋊C8))⋊C3 | 2 |
| C8×(C3⋊D3)⋊C3 | 2 |
| C2×C8×C3⋊D3 | 3 |
| C2×C3×D3×C8 | 3 |